Distributions and Order Parameters

In an isotropic medium, the value of a physical property can be represented by a scalar quantity however, for an anistropic medium the value depends on the measuring angle relative to the direction of alignment designated as the director. The properties of a nematic must be represented by a tensor which in the case of a uniaxial nematic for a property X can be conventionally expressed by Xi=X2=X2 and X-i=X which are, respectively, the values of the property X perpendicular and parallel to the director  

A distribution function /(u, (p)dQ. can be defined which gives the probability of finding rods in a small solid angle dQ=sin u dodtp around the direction (u, tp). As n is defined along the Z-axis,/(u, q ) is independent of q , due to the uniaxial nature of the nematic. Consequently n has complete symmetry around the Z-axis and n = n, therefore /(u) = /(—o). A straightforward summation would be 

This is the second Legrande polynomial. Going to higher order terms gives 

The factor of is introduced to avoid double counting of the interactions. This expression is useful, as for a perfectly aligned nematic u = 0 S = 1, whereas in the random or isotropic liquid cos u=l/3 S = d. Thus nematic liquid crystals have values of S that will range form 0 to 1. 

This mean field theory uses a weak anisotropic potential F f) which is given the form 

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